Vertex Group With Checker Deselect
In the image, you have selected every other face and now have half of the faces selected. Vertex groups can only contain vertices, so for each face, blender is selecting the vertices that make up the four corners of each square. Because your selection of faces touches all the vertices in the cube, Blender adds all the vertices in the cube to the vertex group.

To see exactly what this is doing, try selecting just one face, then switch to vertex select mode to see what vertices selects.

To get every other vertex selected, you can go into vertex select mode and select every other vertex before grouping.